Abstract

We propose and experimentally demonstrate an all-optical (all-fiber) temporal differentiator based on a simple π-phase-shifted fiber Bragg grating operated in reflection. The proposed device can calculate the first time derivative of the complex field of an arbitrary narrowband optical waveform with a very high accuracy and efficiency. Specifically, the experimental fiber grating differentiator reported here offers an operation bandwidth of ≈ 12 GHz. We demonstrate the high performance of this device by processing gigahertz-bandwidth phase and intensity optical temporal variations.

©2007 Optical Society of America

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References

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  1. J. Azaña, C. K. Madsen, K. Takiguchi, and G. Cincontti, eds., “Optical Signal Processing,” J. Ligthwave Technol. 24,2484–2767 (2006).
    [Crossref]
  2. N. Q. Ngo, S. F. Yu, S. C. Tjin, and C.H. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. 230,115–129 (2004).
    [Crossref]
  3. M. Kulishov and J. Azaña, “Long-period fiber gratings as ultrafast optical differentiators,” Opt. Lett. 30,2700–2702 (2005).
    [Crossref] [PubMed]
  4. R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, “Ultrafast all-optical differentiators,” Opt. Express 14,10699–10707 (2006).
    [Crossref] [PubMed]
  5. R. Kashyap, Fiber Bragg Gratings, (Academic Press, San Diego, 1999).
  6. A. Othonos and K. KalliFiber Bragg Gratings. Fundamentals and Applications in Telecommunications and Sensing, (Artech House, Boston, 1999).
  7. H. J. A. Da Silva and J. J. O’Reilly, “Optical pulse modeling with Hermite - Gaussian functions,” Opt. Lett. 14,526–528 (1989).
    [Crossref] [PubMed]
  8. A. Papoulis, The Fourier Integral and its Applications, (McGraw-Hill, New York1987).
  9. M. Yamada and K. Sakuda, “Analysis of almost-periodic distributed feedback slab waveguides via a fundamental transfer matrix approach,” Appl. Opt. 26,3474–3478 (1987).
    [Crossref] [PubMed]
  10. J. E. Bjorkholm, E. H. Turner, and D. B. Pearson, “Conversion of cw light into a train of subnanosecond pulses using frequency modulation and the dispersion of a near-resonant atomic vapor,” Appl. Phys. Lett. 26,564–566 (1975).
    [Crossref]
  11. N. K. Berger, B. Levit, and B. Fischer, “Complete characterization of optical pulses using a chirped fiber Bragg grating,” Opt. Commun. 251,315–321 (2005).
    [Crossref]
  12. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72,156–160 (1982).
    [Crossref]

2006 (2)

J. Azaña, C. K. Madsen, K. Takiguchi, and G. Cincontti, eds., “Optical Signal Processing,” J. Ligthwave Technol. 24,2484–2767 (2006).
[Crossref]

R. Slavík, Y. Park, M. Kulishov, R. Morandotti, and J. Azaña, “Ultrafast all-optical differentiators,” Opt. Express 14,10699–10707 (2006).
[Crossref] [PubMed]

2005 (2)

N. K. Berger, B. Levit, and B. Fischer, “Complete characterization of optical pulses using a chirped fiber Bragg grating,” Opt. Commun. 251,315–321 (2005).
[Crossref]

M. Kulishov and J. Azaña, “Long-period fiber gratings as ultrafast optical differentiators,” Opt. Lett. 30,2700–2702 (2005).
[Crossref] [PubMed]

2004 (1)

N. Q. Ngo, S. F. Yu, S. C. Tjin, and C.H. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. 230,115–129 (2004).
[Crossref]

1989 (1)

1987 (1)

1982 (1)

1975 (1)

J. E. Bjorkholm, E. H. Turner, and D. B. Pearson, “Conversion of cw light into a train of subnanosecond pulses using frequency modulation and the dispersion of a near-resonant atomic vapor,” Appl. Phys. Lett. 26,564–566 (1975).
[Crossref]

Azaña, J.

Berger, N. K.

N. K. Berger, B. Levit, and B. Fischer, “Complete characterization of optical pulses using a chirped fiber Bragg grating,” Opt. Commun. 251,315–321 (2005).
[Crossref]

Bjorkholm, J. E.

J. E. Bjorkholm, E. H. Turner, and D. B. Pearson, “Conversion of cw light into a train of subnanosecond pulses using frequency modulation and the dispersion of a near-resonant atomic vapor,” Appl. Phys. Lett. 26,564–566 (1975).
[Crossref]

Fischer, B.

N. K. Berger, B. Levit, and B. Fischer, “Complete characterization of optical pulses using a chirped fiber Bragg grating,” Opt. Commun. 251,315–321 (2005).
[Crossref]

Ina, H.

Kalli, K.

A. Othonos and K. KalliFiber Bragg Gratings. Fundamentals and Applications in Telecommunications and Sensing, (Artech House, Boston, 1999).

Kam, C.H.

N. Q. Ngo, S. F. Yu, S. C. Tjin, and C.H. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. 230,115–129 (2004).
[Crossref]

Kashyap, R.

R. Kashyap, Fiber Bragg Gratings, (Academic Press, San Diego, 1999).

Kobayashi, S.

Kulishov, M.

Levit, B.

N. K. Berger, B. Levit, and B. Fischer, “Complete characterization of optical pulses using a chirped fiber Bragg grating,” Opt. Commun. 251,315–321 (2005).
[Crossref]

Morandotti, R.

Ngo, N. Q.

N. Q. Ngo, S. F. Yu, S. C. Tjin, and C.H. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. 230,115–129 (2004).
[Crossref]

O’Reilly, J. J.

Othonos, A.

A. Othonos and K. KalliFiber Bragg Gratings. Fundamentals and Applications in Telecommunications and Sensing, (Artech House, Boston, 1999).

Papoulis, A.

A. Papoulis, The Fourier Integral and its Applications, (McGraw-Hill, New York1987).

Park, Y.

Pearson, D. B.

J. E. Bjorkholm, E. H. Turner, and D. B. Pearson, “Conversion of cw light into a train of subnanosecond pulses using frequency modulation and the dispersion of a near-resonant atomic vapor,” Appl. Phys. Lett. 26,564–566 (1975).
[Crossref]

Sakuda, K.

Silva, H. J. A. Da

Slavík, R.

Takeda, M.

Tjin, S. C.

N. Q. Ngo, S. F. Yu, S. C. Tjin, and C.H. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. 230,115–129 (2004).
[Crossref]

Turner, E. H.

J. E. Bjorkholm, E. H. Turner, and D. B. Pearson, “Conversion of cw light into a train of subnanosecond pulses using frequency modulation and the dispersion of a near-resonant atomic vapor,” Appl. Phys. Lett. 26,564–566 (1975).
[Crossref]

Yamada, M.

Yu, S. F.

N. Q. Ngo, S. F. Yu, S. C. Tjin, and C.H. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. 230,115–129 (2004).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

J. E. Bjorkholm, E. H. Turner, and D. B. Pearson, “Conversion of cw light into a train of subnanosecond pulses using frequency modulation and the dispersion of a near-resonant atomic vapor,” Appl. Phys. Lett. 26,564–566 (1975).
[Crossref]

J. Ligthwave Technol. (1)

J. Azaña, C. K. Madsen, K. Takiguchi, and G. Cincontti, eds., “Optical Signal Processing,” J. Ligthwave Technol. 24,2484–2767 (2006).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Commun. (2)

N. K. Berger, B. Levit, and B. Fischer, “Complete characterization of optical pulses using a chirped fiber Bragg grating,” Opt. Commun. 251,315–321 (2005).
[Crossref]

N. Q. Ngo, S. F. Yu, S. C. Tjin, and C.H. Kam, “A new theoretical basis of higher-derivative optical differentiators,” Opt. Commun. 230,115–129 (2004).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Other (3)

R. Kashyap, Fiber Bragg Gratings, (Academic Press, San Diego, 1999).

A. Othonos and K. KalliFiber Bragg Gratings. Fundamentals and Applications in Telecommunications and Sensing, (Artech House, Boston, 1999).

A. Papoulis, The Fourier Integral and its Applications, (McGraw-Hill, New York1987).

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Figures (11)

Fig. 1.
Fig. 1. Numerically simulated (a) field reflectivity and (b) phase of an ideal π phase-shifted FBG around its reflection resonance dip.

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Fig. 2.
Fig. 2. Calculated intensity reflectivity of the non-ideal phase-shifted fiber Bragg grating.

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Fig. 3.
Fig. 3. Calculated (a) reflection spectral dip (logarithmic scale) and (b) reflection spectral phase response of the non-ideal phase-shifted fiber Bragg grating.

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Fig. 4.
Fig. 4. Calculated (a) temporal intensity and (b) discrete energy spectrum of the periodic input optical pulses.

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Fig. 5.
Fig. 5. Calculated squared magnitude of the first time derivative (red curve) of the pulses shown in Fig. 4 and calculated temporal intensity of the pulses reflected from the phase-shifted Bragg grating (blue curve).

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Fig. 6.
Fig. 6. Measured (a) spectral reflectivity and (b) reflection phase response of the fabricated phase-shifted FBG.

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Fig. 7.
Fig. 7. Experimental setup for optical signal differentiation: PC – polarization controller, DCF – dispersion compensating fiber, C – circulator, EDFA – erbium doped fiber amplifier.

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Fig. 8.
Fig. 8. Measured energy spectrum of the sinusoidally phase modulated laser radiation used as an input signal for optical differentiation.

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Fig. 9.
Fig. 9. Numerically calculated squared magnitude of the time derivative of the sinusoidally phase modulated CW light (red curve) and experimentally measured temporal intensity of the output signal after reflection from the phase-shifted fiber Bragg grating (blue curve).

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Fig. 10.
Fig. 10. Experimentally measured (blue curve) and numerically calculated (red curve) temporal intensity of the input pulses formed by sinusoidal phase modulation of CW laser light followed by propagation through a section of dispersion compensating fiber, with the parameters given in the text.

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Fig. 11.
Fig. 11. Numerically calculated squared magnitude of the time derivative (red curve) of the pulses shown in Fig. 10 and experimentally measured intensity of the pulses reflected from the phase-shifted fiber Bragg grating (blue curve).

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Equations (8)

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T ps = T 1 ∙ T φ ∙ T 2
T m 11 = T m 22 * = [ cosh ⁡ ( γ m l m ) + i ( δβ ) l m sin ⁡ h ( γ m l m ) ( γ m l m ) ] exp ⁡ ( i 2 π n eff l m λ B ) ,
T m 12 = T m 21 * = − [ k m l m sinh ⁡ ( γ m l m ) γ m l m ] exp ⁡ ( − i 2 π n eff l m λ B ) ,
T φ 11 = exp ⁡ ( − iφ 2 ) ; T φ 22 = exp ⁡ ( iφ 2 ) ; T φ 12 = T φ 21 = 0
r = T 21 T 11
τ = 1 T 11
r ps = r 1 + r 2 exp ⁡ [ i ( φ + 2 φ 1 τ ) ] 1 + r 1 * r 2 exp ⁡ [ i ( φ + 2 φ 1 τ ) ]
r ps = r [ 1 − exp ⁡ ( 2 i φ 1 τ ) ] 1 − ∣ r ∣ 2 exp ⁡ ( 2 i φ 1 τ )

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